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 SIGMA, 2012, Volume 8, 100, 53 pages (Mi sigma777)

Geometry of Spectral Curves and All Order Dispersive Integrable System

Gaëtan Borota, Bertrand Eynardbc

a Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211 Genève 4, Switzerland
b Institut de Physique Théorique, CEA Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France
c Centre de Recherche Mathématiques de Montréal, Université de Montréal, P.O. Box 6128, Montréal (Québec) H3C 3J7, Canada

Abstract: We propose a definition for a Tau function and a spinor kernel (closely related to Baker–Akhiezer functions), where times parametrize slow (of order $1/N$) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of $1/N$, where the coefficients involve theta functions whose phase is linear in $N$ and therefore features generically fast oscillations when $N$ is large. The large $N$ limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in $1/N$ an isomonodromic problem given by a Lax pair, and the relation between “correlators”, the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.

Keywords: topological recursion; Tau function; Sato formula; Hirota equations; Whitham equations

DOI: https://doi.org/10.3842/SIGMA.2012.100

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ArXiv: 1110.4936
MSC: 14H70; 14H42; 30Fxx
Received: November 14, 2011; in final form December 11, 2012; Published online December 18, 2012
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Citation: Gaëtan Borot, Bertrand Eynard, “Geometry of Spectral Curves and All Order Dispersive Integrable System”, SIGMA, 8 (2012), 100, 53 pp.

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This publication is cited in the following articles:
1. Kalla C., Korotkin D., “Baker-Akhiezer Spinor Kernel and Tau-Functions on Moduli Spaces of Meromorphic Differentials”, Commun. Math. Phys., 331:3 (2014), 1191–1235
2. Alvarez G., Martinez Alonso L., Medina E., “Partition Functions and the Continuum Limit in Penner Matrix Models”, J. Phys. A-Math. Theor., 47:31 (2014), 315205
3. Bergere M., Borot G., Eynard B., “Rational Differential Systems, Loop Equations, and Application To the Qth Reductions of Kp”, Ann. Henri Poincare, 16:12 (2015), 2713–2782
4. Borot, G.; Eynard, B., “All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials”, Quantum Topology, 6:1 (2015), 39-138
5. G. Borot, S. Shadrin, “Blobbed topological recursion: properties and applications”, Math. Proc. Camb. Philos. Soc., 162:1 (2017), 39–87
6. R. Belliard, B. Eynard, O. Marchal, “Integrable differential systems of topological type and reconstruction by the topological recursion”, Ann. Henri Poincare, 18:10 (2017), 3193–3248
7. M. Marino, S. Zakany, “Exact eigenfunctions and the open topological string”, J. Phys. A-Math. Theor., 50:32 (2017), 325401
8. V. Bouchard, B. Eynard, “Reconstructing WKB from topological recursion”, Journal de l'Ecole Polytechnique - Mathematiques, 4 (2017), 845-908
9. R. Belliard, B. Eynard, O. Marchal, “Loop equations from differential systems on curves”, Ann. Henri Poincare, 19:1 (2018), 141–161
10. K. Iwaki, O. Marchal, A. Saenz, “Painlevé equations, topological type property and reconstruction by the topological recursion”, J. Geom. Phys., 124 (2018), 16–54
11. N. Do, P. Norbury, “Topological recursion on the Bessel curve”, Commun. Number Theory Phys., 12:1 (2018), 53–73
12. V. Bouchard, N. K. Chidambaram, T. Dauphinee, “Quantizing weierstrass”, Commun. Number Theory Phys., 12:2 (2018), 253–303
13. Marino M., Zakany S., “Wavefunctions, Integrability, and Open Strings”, J. High Energy Phys., 2019, no. 5, 014
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